Exploration and Exploitation in Parkinson’s Disease: Computational Analyses
Authors
Affiliations
Björn Meder
Health and Medical University, Potsdam, Germany
Martha Sterf
Medical School Berlin, Berlin, Germany
Charley M. Wu
University of Tübingen, Tübingen, Germany
Matthias Guggenmos
Health and Medical University, Potsdam, Germany
Published
August 29, 2025
Code
# Housekeeping: Load packages and helper functions# Housekeepingknitr::opts_chunk$set(echo =TRUE)knitr::opts_chunk$set(message =FALSE)knitr::opts_chunk$set(warning =FALSE)knitr::opts_chunk$set(fig.align='center')options(knitr.kable.NA ='')packages <-c('gridExtra', 'BayesFactor', 'tidyverse', "RColorBrewer", "lme4", "sjPlot", "lsr", "brms", "kableExtra", "afex", "emmeans", "viridis", "ggpubr", "hms", "scales", "cowplot", "waffle", "ggthemes", "parameters", "rstatix", "magick", "grid", "cetcolor", "ggcorrplot")installed <- packages %in%rownames(installed.packages())if (any(!installed)) {install.packages(packages[!installed])}# Load all packageslapply(packages, require, character.only =TRUE)set.seed(0815)# file with various statistical functions, among other things it provides tests for Bayes Factors (BFs)source('statisticalTests.R')# Wrapper for brm models such that it saves the full model the first time it is run, otherwise it loads it from diskrun_model <-function(expr, modelName, path='brm', reuse =TRUE) { path <-paste0(path,'/', modelName, ".brm")if (reuse) { fit <-suppressWarnings(try(readRDS(path), silent =TRUE)) }if (is(fit, "try-error")) { fit <-eval(expr)saveRDS(fit, file = path) } fit}# Setting some plotting paramsw_box <-0.2# width of boxplot, also used for jittering points and lines line_jitter <- w_box /2xAnnotate <--0.3# jitter paramsjit_height <-0.01jit_width <-0.05jit_alpha <-0.6# colors for age groupsgroupcolors <-c("#d95f02", "#1b9e77", "#7570b3")choice3_colors <-c("#e7298a", "#66a61e", "#e6ab02")
1 Preamble
This document provides R code for the statistical analyses and plots of the behavioral data reported in the article
Meder, B., Sterf, M. Wu, C.M, & Guggenmos, M. (2025). Uncertainty-directed and random exploration in Parkinson’s disease. PsyArXiv
All analyses are fully reproducible, with the R code shown alongside the results, and random seeds set to ensure identical outputs across runs. Full session info is provided at the end of the document. All materials, including this document and all data, are available at:
There are two files with behavioral data: data_gridsearch_Parkinson.csv
The behavioral data are stored in
data_gridsearch_parkinson.csv, which contains the behavioral data from rounds 1-9 from the task
data_gridsearch_subjects.csv, which contains participant information.
These files are combined to data frame dat, which includes the following variables:
id: participant id
age is participant age in years
gender: (m)ale, (f)emale, (d)iverse
x and y are the sampled coordinates on the grid
chosen: are the x and y coordinates of the chosen tile
z is the reward obtained from the chosen tile, normalized to the range 0-1. Re-clicked tiles could show small variations in the observed color (i.e., underlying reward) due to normally distributed noise,\(\epsilon∼N(0,1)\).
z_scaled is the observed outcome (reward), scaled in each round to a randomly drawn maximum value in the range of 70% to 90% of the highest reward value
trial is the trial number (0-25), with 0 corresponding to the initially revealed random tile, i.e. trial 1 is the first choice
round is the round number (1 through 10), with 1=practice round (not analyzed) and 10=bonus round (analyzed only for bonus round judgments)
distance is the Manhattan distance between consecutive clicks. NA for trial 0, the initially revealed random tile
type_choice categorizes consecutive clicks as “repeat” (clicking the same tile as in the previous round), “near” (clicking a directly neighboring tile, i.e. distance=1), and “far” (clicking a tile with distance > 1). NA for trial 0, i.e., the initially revealed random tile.
previous_reward is the reward z obtained on the previous step. NA for trial 0, i.e., the initially revealed random tile.
last_ldopa: time of the last L-Dopa dose (HH:MM)
next_ldopa: scheduled time of the next L-Dopa dose (HH:MM)
time_exp: time of the experiment (HH:MM)
time_since_ldopa: time since last L-Dopa (in minutes)
File modelFits.csv contains the results of the computational model simulations (GP-UCB model and lesioned variants).
Complementing the behavioral analyses, we study exploration and exploitation in PD through the lens of a computational model, the Gaussian Process Upper Confidence Bound (GP-UCB) model. This model integrates similarity-based generalization with two distinct exploration mechanisms: directed exploration, which seeks to reduce uncertainty about rewards, and random exploration, which adds stochastic noise to the search process without being directed towards a particular goal (Wu et al., 2018; Wu et al., 2025). In previous research using the same paradigm, this model has provided the best account of human behavior and enabled the decomposition of exploration into distinct mechanisms (Giron et al., 2023; Meder et al., 2021; Schulz et al., 2019; Wu et al., 2018; Wu et al., 2020).
3.1 Gaussian Process Upper Confidence Bound (GP-UCB) Model
The GP-UCB model comprises three components:
a learning model, which uses Bayesian inference to generate predictions about the rewards associated with each option (tile),
a sampling strategy, which uses reward expectations and associated uncertainty to evaluate how promising each option is, and
a choice rule, which converts options’ values into choice probabilities.
Note
Add details
3.1.1 Learning Model
3.1.2 Sampling Strategy
3.1.3 Choice rule
3.1.4 Model parameters
Associated with each model component is a free parameter that we estimate through out-of-sample cross validation. These parameters provide a window into distinct aspects of learning and exploration:
The length-scale parameter \(\lambda\) of the RBF kernel captures how strongly a participant generalizes based on the observed evidence, i.e., the rewards obtained from previous choices.
The uncertainty bonus \(\beta\) represents to the level of directed exploration, i.e., how much expected rewards are inflated through an “uncertainty bonus”.
The temperature parameter \(\tau\) corresponds to the amount of sampling noise, i.e., extent of random exploration.
4 Model comparison
We tested the GP-UCB model in its ability to model learning and predicting each participants’ search and decision-making behavior. To assess the contribution of each component of the model (generalization, uncertainty-directed exploration, and random exploration) we compare the predictive accuracy of the GP-UCB model to model variants where we lesion away each component.
\(\lambda\) lesion model: This model removes the ability to generalize, meaning that all options are learned independently (via Bayesian mean tracker)
\(\beta\) lesion model: No uncertainty-directed exploration (\(\beta=0\)), i.e., options are valued solely based on reward expectations (mean greedy)
\(\tau\) lesion model: Exchanges the softmax choice rule with an \(\epsilon\)-greedy policy as an alternative random exploration mechanism. With probability \(\epsilon\), a random option is selected (each with probability 1/64); with probability 1 − \(\epsilon\), the option with the highest UCB value is chosen. The parameter \(\epsilon\) is estimated for each participant.
All models were fitted using leave-one-round-out cross-validation based on maximum likelihood estimation. Model fits are evaluated using the sum of negative log-likelihoods across all out-of-sample predictions.
Models’ predictive accuracy was assessed using a pseudo-\(R^2\) measure, based on the sum of negative log-likelihoods across all out-of-sample predictions. The summed log loss is compared to a random model, such that \(R^2=0\) corresponds to chance performance and \(R^2=1\) corresponds to theoretically perfect predictions.
Figure 1: Predictive accuracy of GP-UCB model and lesioned variants.
4.1 Model comparison: Control
GP-UCB vs. lambda lesion: \(t(33)=3.0\), \(p=.005\), \(d=0.2\), \(BF=7.5\)
GP-UCB vs. beta lesion: \(t(33)=3.4\), \(p=.002\), \(d=0.2\), \(BF=19\)
GP-UCB vs. tau lesion: \(t(33)=7.7\), \(p<.001\), \(d=0.6\), \(BF>100\)
4.2 Model comparison: PD+
GP-UCB vs. lambda lesion: \(t(32)=3.4\), \(p=.002\), \(d=0.4\), \(BF=20\)
GP-UCB vs. beta lesion: \(t(32)=3.7\), \(p<.001\), \(d=0.4\), \(BF=40\)
GP-UCB vs. tau lesion: \(t(32)=8.5\), \(p<.001\), \(d=0.9\), \(BF>100\)
4.3 Model comparison: PD-
GP-UCB vs. lambda lesion: \(t(30)=3.6\), \(p=.001\), \(d=0.7\), \(BF=27\)
GP-UCB vs. beta lesion: \(t(30)=5.4\), \(p<.001\), \(d=1.1\), \(BF>100\)
GP-UCB vs. tau lesion: \(t(30)=4.9\), \(p<.001\), \(d=1.0\), \(BF>100\)
4.4 Model-based classification of participants
Code
# classify participants according to model R^2df_participant_classification <- modelFits %>%group_by(id) %>%slice_max(order_by = R2, n =1) %>%select(id, group, ModelName, shortname, R2) %>%ungroup() %>%rename(best_ModelName = ModelName,best_shortname = shortname,best_R2 = R2)df_counts <- df_participant_classification %>%count(group, best_shortname)df_percent <- df_counts %>%group_by(group) %>%mutate(total_in_group =sum(n),percent =round((n / total_in_group) *100, 1) ) %>%ungroup()# add most predictive model for each subject to df modelFitsmodelFits <- modelFits %>%left_join(df_participant_classification, by =c("id", "group"))
We classified participants based on which model achieved the highest cross-validated predictive accuracy (highest \(R^2\); ?@fig-participant_classification). In each patient group, the GP-UCB model was the most predictive model for the majority of participants (Control: 55.9%, PD+: 57.6%, PD-: 58.1%).
In total, out of 98 participants, 56 (57.1%) were best described by the GP-UCB model, 22 (22.4%) by the lambda lesion model, 13 (13.3%) by the beta lesion model, and 7 (7.1%) by the tau lesion model. The results suggest that all three components of the GP-UCB model are relevant for predicting participants’ behavior.
To better understand the mechanisms underlying the observed behavioral differences, we analyzed the parameters of the Gaussian Process Upper Confidence Bound (GP-UCB) model (Figure 2).
5.0.1 Generalization \(\lambda\)
The parameter \(\lambda\) represents the length-scale in the RBF kernel, which governs the amount of generalization, i.e., to what extent participants assume a spatial correlation between options (higher \(\lambda\) = stronger generalization). Overall, the amount of generalization was very similar between groups.
Control vs. PD+: \(U=678\), \(p=.145\), \(r_{ au}=.15\), \(BF=.65\)
Control vs. PD-: \(U=731\), \(p=.007\), \(r_{ au}=.28\), \(BF=3.9\)
PD+ vs. PD-: \(U=626\), \(p=.126\), \(r_{ au}=.16\), \(BF=.44\)
5.0.2 Exploration bonus \(\beta\)
The parameter \(\beta\) represents the uncertainty bonus, i.e. how much expected rewards are positively inflated by their uncertainty (higher \(\beta\) = more uncertainty-directed exploration). Controls and PD+ patients on medication did not differ, and both groups had lower beta estimates than the dopamine-depleted patients in the PD− group. These differences suggest that levodopa medication modulated the amount of uncertainty-directed exploration by restoring beta to levels comparable to those observed in controls without PD. This aligns with findings from a restless bandit paradigm, where L-Dopa reduced the amount of directed exploration in healthy volunteers, while the level of random exploration remained unaffected (Chakroun et al., 2020).
Control vs. PD+: \(U=480\), \(p=.315\), \(r_{ au}=-.10\), \(BF=.48\)
Control vs. PD-: \(U=188\), \(p<.001\), \(r_{ au}=-.46\), \(BF=81\)
PD+ vs. PD-: \(U=220\), \(p<.001\), \(r_{ au}=-.41\), \(BF=25\)
5.0.3 Random exploration \(\tau\)
The parameter \(\tau\) represents the amount of decision noise, i.e. stochastic variability in the softmax decision rule (lower \(\tau\) = more decision noise, i.e. more uniform distribution; conversely, \(\tau \rightarrow \infty \quad \Rightarrow \quad \text{argmax (greedy)}\)). There were no group differences in rge temperature paramter \(\tau\), indicating comparable amounts of random exploration regardless of group.
Control vs. PD+: \(U=572\), \(p=.896\), \(r_{ au}=.01\), \(BF=.25\)
Control vs. PD-: \(U=500\), \(p=.730\), \(r_{ au}=-.04\), \(BF=.27\)
PD+ vs. PD-: \(U=470\), \(p=.584\), \(r_{ au}=-.06\), \(BF=.28\)
6 Relations of model parameters to performance
We assessed the correlation (Kendall’s tau, because it’s invariant against log transformation) of GP-UCB parameter estimates with performance (mean reward).
Code
# mean reward per subject across all trials and rounds (practice and bonus round excluded)df_mean_reward_subject <- dat %>%filter(trial !=0& round %in%2:9) %>%# exclude first (randomly revealed) tile and practice round and bonus roundgroup_by(id) %>%summarise(group =first(group),sum_reward =sum(z),mean_reward =mean(z), sd_reward =sd(z)) df_params_performance <- df_gpucb_params %>%left_join(df_mean_reward_subject, by =c("id", "group"))df_params_performance_wide <- df_gpucb_params %>%pivot_wider(names_from = param, values_from = estimate ) %>%left_join(df_mean_reward_subject, by =c("id", "group"))
The amount of generalization was positively related with obtained rewards, showing that participants who successfully learned about the spatially correlation of rewards performed better. The uncertainty bonus \(\beta\) was negatively correlated with performance, demonstrating that an overreliance on exploration impairs efficient reward accumulation. The amount of random temperature \(\tau\) was not related to obtained rewards.
Figure 3: Correlation of GP-UCB parameters with obtained mean reward across all trials and rounds. Each dot is one participant. The insets show the correlations for a restricted parameter range from 0 to 1.
6.1 Generalization \(\lambda\)
Overall, the extent of generalization was positively related to performance, suggesting that participants who stronger generalized obtained more rewards:
Overall: \(r_{ au}=.26\), \(p<.001\), \(BF>100\)
Analysis of parameter estimates on the group level showed that this overall relation was primarily driven by PD+ patients, who showed a strong relation, whereas there was no relation in controls or PD- patients:
Control: \(r_{ au}=.13\), \(p=.288\), \(BF=.39\)
PD+: \(r_{ au}=.45\), \(p<.001\), \(BF>100\)
PD-: \(r_{ au}=-.01\), \(p=.973\), \(BF=.23\)
6.2 Exploration bonus \(\beta\)
The exploration bonus \(\beta\) driving uncertainty-directed correlation was negatively related to performance, suggesting that participants who explore too much at the cost of exploiting known high-value options achieve lower performance:
Overall: \(r_{ au}=-.59\), \(p<.001\), \(BF>100\)
Analysis of parameter estimates on the group level showed that this overall relation was primarily driven by PD+ patients, who showed a strong relation, whereas there was no relation in controls or PD- patients:
Control: \(r_{ au}=-.43\), \(p<.001\), \(BF>100\)
PD+: \(r_{ au}=-.61\), \(p<.001\), \(BF>100\)
PD-: \(r_{ au}=-.60\), \(p<.001\), \(BF>100\)
6.3 Random exploration \(\tau\)
The temperature parameter of the softmax choice rule \(\tau\), representig random exploration, was not related to performance, suggesting that participants who explore too much at the cost of exploiting known high-value options achieve lower performance:
Overall: \(r_{ au}=-.07\), \(p=.308\), \(BF=.22\)
Analysis of parameter estimates on the group level showed that this overall relation was primarily driven by PD+ patients, who showed a strong relation, whereas there was no relation in controls or PD- patients:
Control: \(r_{ au}=-.02\), \(p=.860\), \(BF=.23\)
PD+: \(r_{ au}=.09\), \(p=.451\), \(BF=.30\)
PD-: \(r_{ au}=-.23\), \(p=.077\), \(BF=1.1\)
6.4 Regression: Perfomance and model parameters
We also performed a regression analysis where we included all model paramaters together with group as predictors.
Code
df_params_performance_wide <- df_params_performance %>%filter(ModelName =="GP-UCB") %>%select(id, group, mean_reward, param, estimate_log10) %>%distinct(id, group, param, .keep_all =TRUE) %>%# drop duplicates if anypivot_wider(names_from = param, values_from = estimate_log10) %>%drop_na(beta, tau, lambda) lm_performance_parameters_log <-lm(mean_reward ~ group * (lambda + beta + tau), data = df_params_performance_wide)tab_model(lm_performance_parameters_log)# res.table <- as.data.frame(coef(summary(lm_performance_parameters_log)))# check models# library(performance)# check_model(lm_performance_parameters_log)# # df_params_performance_wide2 <- # df_params_performance %>% # filter(ModelName == "GP-UCB") %>% # select(id, group, mean_reward, param, estimate) %>%# distinct(id, group, param, .keep_all = TRUE) %>% # drop duplicates if any# pivot_wider(names_from = param, values_from = estimate) %>%# drop_na(beta, tau, lambda) # # # lm_performance_parameters <- lm(mean_reward ~ group * (lambda + beta + tau), data = df_params_performance_wide2)# # tab_model(lm_performance_parameters, title = "Regression results: Performance (obtained rewards) as function of group and model parameters.")# # res.table <- as.data.frame(coef(summary(lm_performance_parameters)))# # check_model(lm_performance_parameters)
Table 2: Regression results: Performance (obtained rewards) as function of group and model parameters (log scale).
mean reward
Predictors
Estimates
CI
p
(Intercept)
0.54
0.50 – 0.58
<0.001
group [PD+]
0.10
0.03 – 0.17
0.009
group [Control]
0.08
0.03 – 0.14
0.005
lambda
0.02
-0.20 – 0.24
0.858
beta
-0.02
-0.06 – 0.01
0.206
tau
-0.01
-0.05 – 0.03
0.608
group [PD+] × lambda
0.07
-0.26 – 0.41
0.675
group [Control] × lambda
-0.10
-0.51 – 0.31
0.621
group [PD+] × beta
-0.05
-0.14 – 0.05
0.345
group [Control] × beta
-0.07
-0.15 – 0.01
0.078
group [PD+] × tau
0.03
-0.04 – 0.09
0.447
group [Control] × tau
0.03
-0.06 – 0.12
0.519
Observations
98
R2 / R2 adjusted
0.635 / 0.588
6.5 Model simulations
To evaluate how well different parameter settings balance exploration and exploitation, we conducted simulations with the GP-UCB model. In these simulations, we fixed the value of \(\lambda\) at 1, corresponding to the true amount of correlation in the used environments, and systematically varied the amount of random exploration (\(\tau\)) and the size of the uncertainty bonus (\(\beta\)). For each parameter we defined used equally log-spaced values, and then simulated 100 learners searching for rewards. Environments were sampled (with replacement) from the set of 40 environments used in the empirical study.
Statistical analyses were performed using R. We report both frequentist and Bayesian statistics, using Bayes factors (BF) to quantify the relative evidence of the data in favor of the alternative hypothesis (\(H_1\)) over the null (\(H_0\)). All data and code required for reproducing the statistical analyses and figures are available at ADD GITHUB or OSF LINK.
For parametric group comparisons, we report (paired or independent) Student’s t-tests (two-tailed). For non-parametric comparisons we used the Mann-Whitney U test or Wilcoxon signed-rank test. Bayes factors for the t-tests were computed with the package (Morey & Rouder, 2024), using its default settings. Bayes factor for rank tests were computed following (Doorn et al., 2020).
Linear correlations were assessed using Pearson’s \(r\), with the Bayes factors computed with the BayesFactor package (Morey & Rouder, 2024), using its default settings. Bayes factors for rank correlations quantified with Kendall’s tau were computed using an implementation from Doorn et al. (2018).
7.2 Supplementary computational results
8 Article figure
The following code generates Figure 3 from the article.
GP-UCB model parameters \(\lambda\) (amount of generalization), \(\beta\) (exploration bonus), and \(\tau\) (amount of random exploration) acorss allparticipants
Code
# for now, random intercepts only, Random intercept + random slope not stable# fit models: main effects onlylm_tau_bdi_mmse <-lm(log(tau) ~ group + BDI + MMSE , data = df_params_clinical_indicators)lm_beta_bdi_mmse <-lm(log(beta) ~ group + BDI + MMSE , data = df_params_clinical_indicators)lm_lambda_bdi_mmse <-lm(lambda ~ group + BDI + MMSE , data = df_params_clinical_indicators)tab_model(lm_lambda_bdi_mmse, lm_beta_bdi_mmse, lm_tau_bdi_mmse)
lambda
log(beta)
log(tau)
Predictors
Estimates
CI
p
Estimates
CI
p
Estimates
CI
p
(Intercept)
-0.52
-2.14 – 1.10
0.526
10.81
-1.77 – 23.40
0.091
0.12
-14.16 – 14.40
0.987
group [PD+]
0.01
-0.10 – 0.12
0.893
-1.00
-1.85 – -0.16
0.021
-0.67
-1.63 – 0.29
0.169
group [Control]
0.10
-0.01 – 0.21
0.068
-1.41
-2.24 – -0.57
0.001
-0.62
-1.57 – 0.33
0.199
BDI
0.00
-0.01 – 0.02
0.624
-0.04
-0.13 – 0.06
0.452
-0.01
-0.12 – 0.10
0.817
MMSE
0.04
-0.02 – 0.09
0.198
-0.35
-0.78 – 0.07
0.104
-0.09
-0.57 – 0.40
0.724
Observations
97
97
97
R2 / R2 adjusted
0.060 / 0.020
0.146 / 0.109
0.029 / -0.013
Code
# fit models: main effects + interactions with grouplm_tau_bdi_mmse <-lm(log(tau) ~ group *(BDI + MMSE), data = df_params_clinical_indicators)lm_beta_bdi_mmse <-lm(log(beta) ~ group * (BDI + MMSE), data = df_params_clinical_indicators)lm_lambda_bdi_mmse <-lm(lambda ~ group * (BDI + MMSE), data = df_params_clinical_indicators)tab_model(lm_lambda_bdi_mmse, lm_beta_bdi_mmse, lm_tau_bdi_mmse)
lambda
log(beta)
log(tau)
Predictors
Estimates
CI
p
Estimates
CI
p
Estimates
CI
p
(Intercept)
0.24
-2.85 – 3.32
0.880
3.34
-19.40 – 26.08
0.771
-7.03
-33.87 – 19.82
0.604
group [PD+]
-1.38
-5.74 – 2.99
0.532
2.14
-30.00 – 34.27
0.895
2.68
-35.26 – 40.62
0.889
group [Control]
-0.82
-4.87 – 3.23
0.689
20.50
-9.37 – 50.37
0.176
17.19
-18.08 – 52.45
0.335
BDI
0.00
-0.02 – 0.02
0.925
-0.12
-0.29 – 0.04
0.135
0.00
-0.19 – 0.20
0.973
MMSE
0.01
-0.09 – 0.11
0.844
-0.07
-0.84 – 0.70
0.858
0.16
-0.75 – 1.07
0.732
group [PD+] × BDI
0.01
-0.03 – 0.04
0.726
0.07
-0.17 – 0.30
0.560
-0.07
-0.35 – 0.20
0.598
group [Control] × BDI
-0.00
-0.03 – 0.03
0.984
0.26
0.02 – 0.49
0.031
0.07
-0.21 – 0.34
0.641
group [PD+] × MMSE
0.05
-0.10 – 0.19
0.534
-0.13
-1.21 – 0.95
0.811
-0.10
-1.37 – 1.18
0.881
group [Control] × MMSE
0.03
-0.11 – 0.17
0.646
-0.83
-1.85 – 0.18
0.107
-0.64
-1.84 – 0.57
0.296
Observations
97
97
97
R2 / R2 adjusted
0.066 / -0.019
0.234 / 0.165
0.057 / -0.029
Linear regression with model GP-UCB model paramteres as function of group and depression level (BDI-II) and cognitive functioning (MMSE). All participants
GP-UCB model parameters \(\lambda\) (amount of generalization), \(\beta\) (exploration bonus), and \(\tau\) (amount of random exploration); PD patients only.
Code
# for now, random intercepts only, Random intercept + random slope not stable# fit models: main effects onlylm_tau_bdi_mmse_hy <-lm(log(tau) ~ group + BDI + MMSE + hoehn_yahr,data =subset(df_params_clinical_indicators, group !="Control"))lm_beta_bdi_mmse_hy <-lm(log(beta) ~ group + BDI + MMSE + hoehn_yahr,data =subset(df_params_clinical_indicators, group !="Control"))lm_lambda_bdi_mmse_hy <-lm(lambda ~ group + BDI + MMSE + hoehn_yahr, data =subset(df_params_clinical_indicators, group !="Control"))tab_model(lm_lambda_bdi_mmse_hy, lm_beta_bdi_mmse_hy, lm_tau_bdi_mmse_hy)
lambda
log(beta)
log(tau)
Predictors
Estimates
CI
p
Estimates
CI
p
Estimates
CI
p
(Intercept)
-0.74
-2.88 – 1.39
0.487
7.04
-11.08 – 25.15
0.440
-5.30
-26.64 – 16.04
0.621
group [PD+]
0.01
-0.10 – 0.12
0.853
-1.08
-1.99 – -0.17
0.021
-0.73
-1.80 – 0.35
0.180
BDI
0.00
-0.01 – 0.02
0.688
-0.08
-0.21 – 0.04
0.195
-0.03
-0.18 – 0.12
0.676
MMSE
0.04
-0.03 – 0.11
0.256
-0.19
-0.79 – 0.42
0.542
0.11
-0.61 – 0.82
0.765
hoehn yahr
0.05
-0.04 – 0.13
0.282
-0.35
-1.07 – 0.36
0.329
0.01
-0.83 – 0.85
0.980
Observations
64
64
64
R2 / R2 adjusted
0.037 / -0.028
0.137 / 0.079
0.036 / -0.029
Code
# fit models: main effects and interactionslm_tau_bdi_mmse_hy <-lm(log(tau) ~ group * (BDI + MMSE + hoehn_yahr),data =subset(df_params_clinical_indicators, group !="Control"))lm_beta_bdi_mmse_hy <-lm(log(beta) ~ group * (BDI + MMSE + hoehn_yahr),data =subset(df_params_clinical_indicators, group !="Control"))lm_lambda_bdi_mmse_hy <-lm(lambda ~ group * (BDI + MMSE + hoehn_yahr), data =subset(df_params_clinical_indicators, group !="Control"))tab_model(lm_lambda_bdi_mmse_hy, lm_beta_bdi_mmse_hy, lm_tau_bdi_mmse_hy)
lambda
log(beta)
log(tau)
Predictors
Estimates
CI
p
Estimates
CI
p
Estimates
CI
p
(Intercept)
-0.69
-3.62 – 2.24
0.640
6.50
-19.80 – 32.80
0.622
-10.59
-41.20 – 20.03
0.491
group [PD+]
-0.11
-4.25 – 4.04
0.959
0.07
-37.12 – 37.25
0.997
9.70
-33.59 – 52.99
0.655
BDI
-0.00
-0.02 – 0.02
0.782
-0.11
-0.29 – 0.07
0.237
-0.01
-0.23 – 0.20
0.913
MMSE
0.03
-0.06 – 0.13
0.501
-0.15
-1.02 – 0.73
0.738
0.24
-0.78 – 1.27
0.633
hoehn yahr
0.15
0.04 – 0.27
0.011
-0.53
-1.58 – 0.52
0.320
0.59
-0.63 – 1.82
0.335
group [PD+] × BDI
0.01
-0.02 – 0.04
0.494
0.06
-0.21 – 0.32
0.664
-0.05
-0.36 – 0.26
0.734
group [PD+] × MMSE
0.01
-0.12 – 0.15
0.829
-0.08
-1.32 – 1.15
0.897
-0.27
-1.71 – 1.17
0.709
group [PD+] × hoehn yahr
-0.21
-0.37 – -0.05
0.013
0.36
-1.11 – 1.82
0.628
-1.14
-2.84 – 0.56
0.185
Observations
64
64
64
R2 / R2 adjusted
0.145 / 0.039
0.148 / 0.041
0.070 / -0.046
Linear regression with model GP-UCB model paramteres as function of group and depression level (BDI-II) and cognitive functioning (MMSE). PD patients only.
10 Model params of participants best explained by GP-UCB model
For this analysis we only consider participants who were best explained by the GP-UCB model. The results are consistent with the same analyses performed with the full sample above: no substantial differences in amount of generalization \(\lambda\), marked differences in terms of the exploration bonus \(\beta\), and no differences in terms of random exploration \(\tau\). The only difference is that we found a difference between the control and off-medication group in the extent of generalization when using the full sample, whereas we found no difference when only considering the subset of participants best accounted for by the GP-UCB model.
10.0.1 Generalization \(\lambda\)
The parameter \(\lambda\) represents the length-scale in the RBF kernel, which governs the amount of generalization, i.e., to what extent participants assume a spatial correlation between options (higher \(\lambda\) = stronger generalization). Overall, the amount of generalization was very similar between groups.
Control vs. PD+: \(U=210\), \(p=.402\), \(r_{ au}=.12\), \(BF=.44\)
Control vs. PD-: \(U=219\), \(p=.150\), \(r_{ au}=.20\), \(BF=.63\)
PD+ vs. PD-: \(U=191\), \(p=.558\), \(r_{ au}=.08\), \(BF=.34\)
10.0.2 Exploration bonus \(\beta\)
The parameter \(\beta\) represents the uncertainty bonus, i.e. how much expected rewards are positively inflated by their uncertainty (higher \(\beta\) = more uncertainty-directed exploration). Controls and PD+ patients on medication did not differ, and both groups had lower beta estimates than the dopamine-depleted patients in the PD− group. These differences suggest that levodopa medication modulated the amount of uncertainty-directed exploration by restoring beta to levels comparable to those observed in controls without PD. This aligns with findings from a restless bandit paradigm, where L-Dopa reduced the amount of directed exploration in healthy volunteers, while the level of random exploration remained unaffected (Chakroun et al., 2020).
Control vs. PD+: \(U=182\), \(p=.977\), \(r_{ au}=.01\), \(BF=.32\)
Control vs. PD-: \(U=55\), \(p<.001\), \(r_{ au}=-.49\), \(BF=14\)
PD+ vs. PD-: \(U=54\), \(p<.001\), \(r_{ au}=-.49\), \(BF=31\)
10.0.3 Random exploration \(\tau\)
The parameter \(\tau\) represents the amount of decision noise, i.e. stochastic variability in the softmax decision rule (lower \(\tau\) = more decision noise, i.e. more uniform distribution; conversely, \(\tau \rightarrow \infty \quad \Rightarrow \quad \text{argmax (greedy)}\)). There were no group differences in rge temperature paramter \(\tau\), indicating comparable amounts of random exploration regardless of group.
Control vs. PD+: \(U=193\), \(p=.729\), \(r_{ au}=.05\), \(BF=.35\)
Control vs. PD-: \(U=162\), \(p=.799\), \(r_{ au}=-.04\), \(BF=.33\)
PD+ vs. PD-: \(U=140\), \(p=.358\), \(r_{ au}=-.13\), \(BF=.44\)
Figure 6: Parameter estimates of GP-UCB model, estimated through leave-one-round-out cross validation. Each dot is one participant. Only participants are included who were best described by the GP-UCB model.
Chakroun, K., Mathar, D., Wiehler, A., Ganzer, F., & Peters, J. (2020). Dopaminergic modulation of the exploration/exploitation trade-off in human decision-making. eLife, 9, e51260.
Doorn, J. van, Ly, A., Marsman, M., & Wagenmakers, E.-J. (2018). Bayesian inference for kendall’s rank correlation coefficient. The American Statistician, 72, 303–308.
Doorn, J. van, Ly, A., Marsman, M., & Wagenmakers, E.-J. (2020). Bayesian rank-based hypothesis testing for the rank sum test, the signed rank test, and spearman’s \(\rho\). Journal of Applied Statistics, 47(16), 2984–3006.
Giron, A. P., Ciranka, S., Schulz, E., Bos, W. van den, Ruggeri, A., Meder, B., & Wu, C. M. (2023). Developmental changes in exploration resemble stochastic optimization. Nature Human Behaviour, 7(11), 1955–1967. https://doi.org/https://doi.org/10.1038/s41562-023-01662-1
Meder, B., Wu, C. M., Schulz, E., & Ruggeri, A. (2021). Development of directed and random exploration in children. Developmental Science, 24(4), e13095. https://doi.org/https://doi.org/10.1111/desc.13095
Schulz, E., Wu, C. M., Ruggeri, A., & Meder, B. (2019). Searching for rewards like a child means less generalization and more directed exploration. Psychological Science, 30(11), 1561–1572. https://doi.org/10.1177/0956797619863663
Wu, C. M., Meder, B., & Schulz, E. (2025). Unifying principles of generalization: Past, present, and future. Annual Review of Psychology, 76, 275–302. https://doi.org/https://doi.org/10.1146/annurev-psych-021524-110810
Wu, C. M., Schulz, E., Garvert, M. M., Meder, B., & Schuck, N. W. (2020). Similarities and differences in spatial and non-spatial cognitive maps. PLOS Computational Biology, 16(9), e1008149. https://doi.org/10.1371/journal.pcbi.1008149
Wu, C. M., Schulz, E., Speekenbrink, M., Nelson, J. D., & Meder, B. (2018). Generalization guides human exploration in vast decision spaces. Nature Human Behaviour, 2, 915–924. https://doi.org/10.1038/s41562-018-0467-4